![]() ![]() ![]() We always knew tangent was odd, but now we have proof. tan (- a) = -tan a, so it is an odd function. However, checking out the function values for a and - a, we see that works the same as sine. It has a different distribution of signs than sine or cosine. We're on another tangent for tangent (and cotangent). ![]() Knowing this, we can see that sin(- a) = -sin a, which is just like the f( -x) = -f( x) of an odd function. Going in the opposite direction, - a is in negative territory. We can see that angle a is in the positive territory. It has a handy formula too:įor sine (and its reciprocal, cosecant), we have this breakdown of signs: This is symmetry over the origin, where we are flipping over both the x- and y-axes. This can be computed using calculus using the formula for arc length in polar coordinates, (2) but since, this becomes simply (3) The circumference -to- diameter ratio for a circle is constant as the size of the circle is changed (as it must be since scaling a plane figure by a factor increases its perimeter by ), and also scales by. However, if we compare any x and - x, their y values are opposite: y and - y. If we take this graph and fold it over the y-axis, we can see that it doesn't match up, so it is not symmetric over the y-axis it is not an even function. Turns out that y = sin x is an odd function. Odd functions have symmetric over the origin. If there exists something called "even" functions, is it really any surprise that there are odd functions as well? Pick your jaw up off the floor it will get dirt on your chin. That's totally what we wanted, and we got it. Def: If the terminal side of an angle is in standard position and intersects the unit circle at P(x,y) then x cos and y sin Trig functions defined. Plus, Quadrants I and IV are both positive for cosine. The absolute value of cos a and cos (- a) will be the same, because they have the same reference angle. Here are the signs for cosine (and secant) for the angles a and - a: There's so much yarn, how are you going to get out, cat? Oh, it also tells us which functions are positive in each quadrant:Ī(ll) S(ine) T(angent) C(osine). Some people say All Students Take Calculus, but we like to think about it as A Small Tangled Cat. To do this, we remember the memory device, ASTC. We can check that cosine fits the equation by looking at the unit circle. That, and "Feed me, Seymour," but we're not listening to that old song and dance. That's an even function's symmetry, and that's exactly what the equation says. Compare every - x value to x: they have the same value of y. Take off the blindfold and take another look at the graph. We can instead see if the function fits this equation: We don't have to look at a graph to show that a function is even. Any function that is symmetric over the y-axis is an even function. We could fold the whole graph over x = 0 and everything would match up on the other side. There are two types of symmetry when we look at trig functions. Well, functions can have symmetry too, and trig functions are like the sumo symmetry champs. It is commonly used in mathematics, particularly in trigonometry, to define the values of trigonometric functions such as sine, cosine, and tangent at various. That's understandable it's easy to fold shapes in half in different ways, to see if they match up. We tend to think about symmetry in terms of geometry more than anything else. ![]()
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